I couldn't understand the last part of the argument. Why is the length of a side in a 4 sided stuff proportional to angle?

Question

Why is DE =3?

I don't quite get it. What theorem did he use?

https://www.youtube.com/watch?v=Dk9mfrIL7FQ

I also don't understand why all four corners of a four sided object is in a circle if the sum of their opposing angle is 180 degree

Answer 1

You can prove that if two chords of a circle subtend equal angles on the circle's circumference, then they are of equal length. You would typically first prove that the chords are equal in length if they subtend equal angles at the centre, and then use the relationship between the angles at the centre and on the circumference to prove the required result.

For your second question, there is a well-known theorem in geometry that states:

• A convex quadrilateral is cyclic (i.e. there exists a circle touching all four vertices) if and only if the opposing vertices are supplementary.

There is a very brief version of a proof in this document, or a more detailed version is in the section on "Cyclic Quadrilaterals" on this webpage. As the second source notes, the forward proof (that the opposite angles of a cyclic quadrilateral are supplementary) is fairly straightforward, but the other direction (that a quadrilateral whose opposite angles are supplementary must be cyclic) is considered "more difficult than other Year 10 material" in an Australian high school curriculum.